Ideas of Quantum Chemistry P17 docx
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126
3. Beyond the Schrödinger Equation
=|N|
2
(1s)
ˆ
p
z
1
r
ˆ
p
z
(1s)
+
(1s)
ˆ
p
x
−i
ˆ
p
y
1
r
ˆ
p
x
+i
ˆ
p
y
(1s)
=|N|
2
(1s)
ˆ
p
z
1
r
ˆ
p
z
(1s)
+
(1s)
ˆ
p
x
−i
ˆ
p
y
1
r
ˆ
p
x
+i
ˆ
p
y
(1s)
+
(1s)
1
r
ˆ
p
z
ˆ
p
z
(1s)
+
(1s)
1
r
ˆ
p
x
−i
ˆ
p
y
ˆ
p
x
+i
ˆ
p
y
(1s)
(3.63)
In the second row, the scalar product of spinors is used, in the third row, the
Hermitian character of the operator
ˆ
p.Further,
φ
1
r
φ
=|N|
2
(1s)
ˆ
p
z
1
r
ˆ
p
z
(1s)
+
(1s)
1
r
ˆ
p
2
x
+
ˆ
p
2
y
+
ˆ
p
2
z
(1s)
+
(1s)
ˆ
p
x
−i
ˆ
p
y
1
r
ˆ
p
x
+i
ˆ
p
y
(1s)
=|N|
2
(1s)
ˆ
p
z
1
r
ˆ
p
z
(1s)
−
(1s)
1
r
(1s)
+
(1s)
ˆ
p
x
1
r
ˆ
p
x
(1s)
+
(1s)
ˆ
p
y
1
r
ˆ
p
y
(1s)
−i
(1s)
ˆ
p
y
1
r
ˆ
p
x
(1s)
+i
(1s)
ˆ
p
x
1
r
ˆ
p
y
(1s)
(3.64)
We used the atomic units and therefore
ˆ
p
2
=−, and the momentum operator
is equal to −i∇. The two integrals at the end cancel each other, because each of
the integrals does not change when the variables are interchanged: x ↔y.
Finally, we obtain the following formula
φ
1
r
φ
=−|N|
2
1s
1
r
(1s)
+
1s
∇
1
r
∇(1s)
=−ζ
−2
−3ζ
3
+2ζ
3
=ζ
where the equality follows from a direct calculation of the two integrals.
33
The next matrix element to calculate is equal to φ|c(σ ·π)ψ. We proceed as
follows (please recall kinetic balancing and we also use Appendix H, p. 969):
φ|c(σ ·π)ψ=Nc
(σ ·π)
1s
0
(σ ·π)
1s
0
33
In the first integral we have the same situation as a while before. In the second integral we write the
nabla operator in Cartesian coordinates, obtain a scalar product of two gradients, then we get three
integrals equal to one another (they contain x y z), and it is sufficient to calculate one of them by
spherical coordinates by formula (H.2) in Appendix H, p. 969.
3.4 The hydrogen-like atom in Dirac theory
127
= Nc
ˆ
p
z
(1s)
(
ˆ
p
x
+i
ˆ
p
y
)(1s)
ˆ
p
z
(1s)
(
ˆ
p
x
+i
ˆ
p
y
)(1s)
= Nc
ˆ
p
z
(1s)
ˆ
p
z
(1s)
+
(
ˆ
p
x
+i
ˆ
p
y
)(1s)
(
ˆ
p
x
+i
ˆ
p
y
)(1s)
= Nc
1s
ˆ
p
2
(1s)
=
1
ζ
cζ
2
=cζ
Thelastmatrixelementreadsas
ψ|c(σ ·π)φ=Nc
1s
0
(σ ·π)
2
1s
0
= Nc
1s
0
ˆ
p
2
0
0
ˆ
p
2
1s
0
=Nc
1s
ˆ
p
2
1s
=c
1
ζ
ζ
2
=cζ
Dirac’s secular determinant
We have all the integrals needed and may now write the secular determinant cor-
responding to the matrix form of the Dirac equation:
ψ|V ψ−ε ψ|c(σ ·π)φ
φ|c(σ ·π)ψφ|(V −2c
2
))φ−ε
=0
and after inserting the calculated integrals
−Zζ −εcζ
cζ −Zζ −2c
2
−ε
=0
Expanding the determinant gives the equation for the energy ε
ε
2
+ε
2Zζ +2c
2
+
Zζ
Zζ +2c
2
−c
2
ζ
2
=0
Hence, we get two solutions
ε
±
=−
c
2
+Zζ
±
c
4
+ζ
2
c
2
Note that the square root is of the order of c
2
(in a.u.), and with the (unit) mass
of the electron m
0
,itisoftheorderofm
0
c
2
. Therefore, the minus sign before the
square root corresponds to a solution with energy of the order of −2m
0
c
2
,while
the plus sign corresponds to energy of the order of zero. Let us recall that we have
shifted the energy scale in the Dirac equation and the last solution ε
+
(hereafter
denoted by ε) is to be compared to the energy of the non-relativistic hydrogen-like
atom
ε =−
c
2
+Zζ
+
c
4
+ζ
2
c
2
=−
c
2
+Zζ
+c
2
1 +
ζ
2
c
2
128
3. Beyond the Schrödinger Equation
=−
c
2
+Zζ
+c
2
1 +
ζ
2
2c
2
−
ζ
4
8c
4
+
=−Zζ +
ζ
2
2
+
−
ζ
4
8c
2
+
(3.65)
Non-relativistic solution
If c →∞, i.e. we approach the non-relativistic limit, then ε= −Zζ +
ζ
2
2
. Mini-
mization of this energy with respect to ζ gives its optimum value ζ
nonrel
opt
= Z.In
this way one recovers the result known from non-relativistic quantum mechanics
(Appendix H) obtained in the variational approach to the hydrogen atom with the
1s orbital as a trial function.
3.4.2 RELATIVISTIC CONTRACTION OF ORBITALS
Minimizing the relativistic energy equation (3.65) leads to an equation for opti-
mum ζ ≡ζ
rel
opt
:
dε
dζ
=0 =−Z +
1
2
c
4
+ζ
2
c
2
−
1
2
2ζc
2
=−Z +
c
4
+ζ
2
c
2
−
1
2
ζc
2
giving
ζ
rel
opt
=
Z
1 −
Z
2
c
2
The result differs remarkably from the non-relativistic value ζ
nonrel
opt
=Z,butap-
proaches the non-relativistic value when c →∞. Note than the difference between
the two values increases with atomic number Z, and that the relativistic exponent
is always larger that its non-relativistic counter-part. This means that the relativistic
orbital decays faster with the electron–nucleus distance and therefore
the relativistic orbital 1s is smaller (contraction) than the corresponding
non-relativistic one.
Let us see how it is for the hydrogen atom. In that case ζ
rel
opt
= 10000266
as compared to ζ
nonrel
opt
= Z
H
= 1. And what about 1s orbital of gold? For gold
ζ
rel
opt
=9668, while ζ
nonrel
opt
=Z
Au
=79! Since for a heavy atom, the effective expo-
nent of the atomic orbitals decreases when moving from the low-energy compact
1s orbital to higher-energy outer orbitals, this means that the most important rel-
ativistic orbital contraction occurs for the inner shells. The chemical properties of
an atom depend on what happens to its outer shells (valence shell). Therefore, we
3.5 Larger systems
129
may conclude that the relativistic corrections are expected to play a secondary role
in chemistry.
34
If we insert ζ
rel
opt
in eq. (3.65) we obtain the minimum value of ε
ε
min
=−
c
2
+Zζ
+
c
4
+ζ
2
(3.66)
Since Z
2
/c
2
is small with respect to 1, we may expand the square root in the
Taylor series,
√
1 −x =1 −
1
2
x −
1
8
x
2
−···. We obtain
ε
min
=−c
2
+c
2
1 −
1
2
Z
2
c
2
−
1
8
Z
2
c
2
2
−···
=−
Z
2
2
1 +
Z
2c
2
+···
(3.67)
In the case of the hydrogen atom (Z =1) we have
ε
min
=−
1
2
1 +
1
2c
2
+···
(3.68)
where the first two terms shown give Darwin’s exact result
35
(discussed earlier).
Inserting c = 137036 a.u. we obtain the hydrogen atom ground-state energy ε =
−05000067 a.u., which agrees with Darwin’s result.
3.5 LARGER SYSTEMS
The Dirac equation represents an approximation
36
and refers to a single particle.
What happens with larger systems? Nobody knows, but the first idea is to con-
struct the total Hamiltonian as a sum of the Dirac Hamiltonians for individual par-
ticles plus their Coulombic interaction (the Dirac–Coulomb approximation). This
Dirac–Coulomb
approximation
is practised routinely nowadays for atoms and molecules. Most often we use the
mean-field approximation (see Chapter 8) with the modification that each of the
one-electron functions represents a four-component bispinor. Another approach
is extremely pragmatic, maybe too pragmatic: we perform the non-relativistic cal-
culations with a pseudopotential that mimics what is supposed to happen in a rel-
ativistic case.
34
We have to remember, however, that the relativistic effects also propagate from the inner shells to
the valence shell through the orthogonalization condition, that has to be fulfilled after the relativistic
contraction. This is why the gold valence orbital 6s shrinks, which has an immediate consequence in the
relativistic shortening of the bond length in Au
2
, which we cited at the beginning of this chapter.
35
I.e. the exact solution to the Dirac equation for the electron in the external electric field produced
by the proton.
36
Yet it is strictly invariant with respect to the Lorentz transformation.
130
3. Beyond the Schrödinger Equation
3.6 BEYOND THE DIRAC EQUATION
How reliable is the presented relativistic quantum theory? The Dirac or Klein–
Gordon equations, as is usual in physics, describe only some aspects of reality.
The fact that both equations are invariant with respect to the Lorentz transforma-
tion indicates only that the space-time symmetry properties are described correctly.
The physical machinery represented by these equations is not so bad, since several
predictions have been successfully made (antimatter, electron spin, energy levels
of the hydrogen atom). Yet, in the latter case an assumption of the external field
V =−
Ze
2
r
is a positively desperate step, which in fact is unacceptable in a fair rel-
ativistic theory for the proton and the electron (and not only of the electron in
the external field of the nucleus). Indeed, the proton and the electron move. At a
given time their distance is equal to r, but such a distance might be inserted into the
Coulombic law if the speed of light were infinite, because the two particles would
feel their positions instantaneously. Since, however, any perturbation by a posi-
tional change of a particle needs time to travel to the other particle, we have to use
another distance somehow taking this into account (Fig. 3.3). The same pertains,
of course, to any pair of particles in a many-body system (the so-called retarded
retarded
potential
potential).
There is certainly a need for a more accurate theory.
3.6.1 THE BREIT EQUATION
Breit constructed a many-electron relativistic theory that takes into account such
a retarded potential in an approximate way. Breit explicitly considered only the
electrons of an atom, nucleus of which (similar to Dirac theory) created only an
external field for the electrons. This ambitious project was only partly success-
Fig. 3.3. Retardation of the interaction. The dis-
tance r
12
of two particles in the interaction po-
tential (as in Coulomb’s law) is bound to repre-
sent an approximation, because we assume an in-
stantaneous interaction. However, when the two
particles catch sight of each other (which takes
time) they are already somewhere else.
3.6 Beyond the Dirac equation
131
ful, because the resulting theory turned
out to be approximate not only from the
point of view of quantum theory (some
interactions not taken into account) but
also from the point of view of relativity
theory (an approximate Lorentz trans-
formation invariance).
For two electrons the Breit equation
has the form (r
12
stands for the distance
between electron 1 and electron 2)
Gregory Breit (1899–1981),
American physicist, professor
at the universities New York,
Wisconsin, Yale, Buffalo. Breit
with Eugene Wigner intro-
duced the resonance states
of particles, and with Condon
created the proton–proton
scattering theory.
ˆ
H(1) +
ˆ
H(2) +
1
r
12
−
1
2r
12
α(1)α(2) +
[
α(1) ·r
12
][
α(2) ·r
12
]
r
2
12
=E
(3.69)
where (cf. eq. (3.54) with E replaced by the Hamiltonian)
ˆ
H(i) =q
i
φ(r
i
) +cα(i)π(i) +α
0
(i)m
0
c
2
=−eφ(r
i
) +cα(i)π(i) +α
0
(i)m
0
c
2
is the Dirac Hamiltonian for electron i pointed by vector r
i
, whereas the Dirac ma-
trices for electron i: α(i) =[α
x
(i) α
y
(i) α
z
(i)] and the corresponding operators
π
μ
(i) have been defined on p. 114, φ(r
i
) represents the scalar potential calculated
at r
i
. The wavefunction represents a 16-component spinor (here represented
by a square matrix of rank 4), because for each electron we would have the usual
Dirac bispinor (four component) and the two-electron wavefunction depends on
the Cartesian product of the components.
37
The Breit Hamiltonian (in our example, for two electrons in an electromagnetic
field) can be approximated by the following useful formula
38
known as the Breit–
Pauli Hamiltonian
Breit–Pauli
Hamiltonian
ˆ
H(1 2) =
ˆ
H
0
+
ˆ
H
1
+···+
ˆ
H
6
(3.70)
where:
•
ˆ
H
0
=
ˆ
p
2
1
2m
0
+
ˆ
p
2
2
2m
0
+V represents the familiar non-relativistic Hamiltonian.
•
ˆ
H
1
=−
1
8m
3
0
c
2
(
ˆ
p
4
1
+
ˆ
p
4
2
) comes from the velocity dependence of mass, more pre-
cisely from the Taylor expansion of eq. (3.38), p. 109, for small velocities.
•
ˆ
H
2
=−
e
2
2(m
0
c)
2
1
r
12
[
ˆ
p
1
·
ˆ
p
2
+
r
12
·(r
12
·
ˆ
p
1
)
ˆ
p
2
r
2
12
]stands for the correction
39
that accounts
in part for the above mentioned retardation. Alternatively, the term may be
viewed as the interaction energy of two magnetic dipoles, each resulting from
the orbital motion of an electron (orbit–orbit term).
orbit–orbit term
37
In the Breit equation (3.69) the operators in {}act either by multiplying the 4 ×4matrix by a
function (i.e. each element of the matrix) or by a 4 ×4matrixresultingfromα matrices.
38
H.A. Bethe, E.E. Salpeter, “Quantum Mechanics of One- and Two-Electron Atoms”, Springer, 1977,
p. 181.
39
For non-commuting operators
ˆ
a(
ˆ
a ·
ˆ
b)
ˆ
c =
3
ij=1
ˆ
a
i
ˆ
a
j
ˆ
b
j
ˆ
c
i
.
132
3. Beyond the Schrödinger Equation
•
ˆ
H
3
=
μ
B
m
0
c
{[E(r
1
) ×
ˆ
p
1
+
2e
r
3
12
r
12
×
ˆ
p
2
]·s
1
+[E(r
2
) ×
ˆ
p
2
+
2e
r
3
12
r
21
×
ˆ
p
1
]·s
2
} is
the interaction energy of the electronic magnetic moments (resulting from the
above mentioned orbital motion) with the spin magnetic dipole moments (spin–
spin–orbit
coupling
orbit coupling), μ
B
stands for the Bohr magneton, and E denotes the electric
field vector. Since we have two orbital magnetic dipole moments and two spin
orbital dipole moments, there are four spin–orbit interactions. The first term
in square brackets stands for the spin–orbit coupling of the same electron, while
the second term represents the coupling of the spin of one particle with the orbit
of the second.
•
ˆ
H
4
=
ie
¯
h
(2m
0
c)
2
[
ˆ
p
1
·E(r
1
) +
ˆ
p
2
·E(r
2
)] is a non-classical term peculiar to the Dirac
theory (also present in the one-electron Dirac Hamiltonian) called the Darwin
Darwin term
term.
•
ˆ
H
5
= 4μ
2
B
{−
8π
3
(s
1
· s
2
)δ(r
12
) +
1
r
3
12
[s
1
· s
2
−
(s
1
·r
12
)(s
2
·r
12
)
r
2
12
]} corresponds to the
spin dipole moment interactions of the two electrons (spin–spin term). The first
spin–spin
term is known as the Fermi contact term, since it is non-zero only when the two
Fermi contact
term
electrons touch one another (see Appendix E, p. 951), whereas the second term
represents the classical dipole–dipole interaction of the two electronic spins (cf.
the multipole expansion in Appendix X, p. 1038 and Chapter 13), i.e. the in-
teraction of the two spin magnetic moments of the electrons (with the factor 2,
according to eq. (3.62), p. 122).
•
ˆ
H
6
= 2μ
B
[H(r
1
) · s
1
+ H(r
2
) · s
2
]+
e
m
0
c
[A(r
1
) ·
ˆ
p
1
+ A(r
2
) ·
ˆ
p
2
] is known as
the Zeeman interaction, i.e. the interaction of the spin (the first two terms) and
Zeeman term
the orbital (the second two terms) electronic magnetic dipole moments with the
external magnetic field H (cf. eq. (3.62)).
The terms listed above are of prime importance in the theory of the interaction
of matter with the electromagnetic field (e.g., in nuclear magnetic resonance).
3.6.2 A FEW WORDS ABOUT QUANTUM ELECTRODYNAMICS (QED)
The Dirac and Breit equations do not account for several subtle effects.
40
They are
predicted by quantum electrodynamics, a many-particle theory.
Willis Eugene Lamb (b. 1913), American physi-
cist, professor at Columbia, Stanford, Oxford,
Yale and Tucson universities. He received the
Nobel Prize in 1955 “for his discoveries con-
cerning the fine structure of the hydrogen
spectrum”.
40
For example, an effect observed in spectroscopy for the first time by Willis Lamb.
3.6 Beyond the Dirac equation
133
The QED energy may be conveniently developed in a series of
1
c
:
• in zero order we have the non-relativistic approximation (solution to the
Schrödinger equation);
• there are no first order terms;
• the second order contains the Breit corrections;
• the third and further orders are called the radiative corrections.
radiative
corrections
Radiative corrections
The radiative corrections include:
• Interaction with the vacuum (Fig. 3.4.a). According to modern physics the per-
fect vacuum does not just represent nothing. The electric field of the vacuum
itself fluctuates about zero and these instantaneous fluctuations influence the
motion of any charged particle. When a strong electric field operates in a vac-
uum, the latter undergoes a polarization (vacuum polarization), which means a
vacuum
polarization
spontaneous creation of matter, more specifically, of particle-antiparticle pairs.
Fig. 3.4. (a) The electric field close to the proton (composed of three quarks) is so strong that it creates
matter and antimatter (shown as electron–positron pairs). The three quarks visible in scattering exper-
iments represent the valence quarks. (b) One of the radiative effects in the QED correction of the
c
−3
order (see Table 3.1). The pictures show the sequence of the events from left to the right. A pho-
ton (wavy line on the left) polarizes the vacuum and an electron–positron pair (solid lines) is created,
and the photon vanishes. Then the created particles annihilate each other and a photon is created.
(c) A similar event (of the c
−4
order in QED), but during the existence of the electron–positron pair
the two particles interact by exchange of a photon. (d) An electron (horizontal solid line) emits a pho-
ton, which creates an electron–positron pair, that annihilates producing another photon. Meanwhile
the first electron emits a photon, then first absorbs the photon from the annihilation, and afterwards
the photon emitted by itself earlier. This effect is of the order c
−5
in QED.
134
3. Beyond the Schrödinger Equation
The probability of this event (per unit volume and time) depends
41
(Fig. 3.4.a–
d) on the particle mass m and charge q:
w =
E
2
cπ
2
∞
n=1
1
n
2
exp
−
nπm
2
|qE|
(3.71)
where E is the electric field intensity. The creation of such pairs in a static elec-
tricfieldhasneveryetbeenobserved,becausewecannotyetprovidesufficientE.
Even for the electron on the first Bohr orbit, the |qE| is small compared to m
2
(however, for smaller distances the exponent may be much smaller).
creation of
matter
• Interaction with virtual photons. The electric field influences the motion of elec-
tron. What about its own electric field? Does it influence its motion as well? The
latter effect is usually modelled by allowing the electron to emit photons and
then to absorb them (“virtual photons”)
42
(Fig. 3.4.d).
The QED calculations performed to date have been focused on the energy. The
first calculations of atomic susceptibilities (helium) within an accuracy including
the c
−2
terms were carried out independently
43
by Pachucki and Sapirstein
44
and
by Cencek and coworkers,
45
and with accuracy up to c
−3
(with estimation of the
c
−4
term) by Łach and coworkers (see Table 3.1). To get a flavour of what subtle
effects may be computed nowadays, Table 3.1 shows the components of the first
ionization energy and of the dipole polarizability (see Chapter 12) of the helium
atom.
Comments to Table 3.1
•
ˆ
H
0
denotes the result obtained from an accurate solution of the Schrödinger
equation (i.e. the non-relativistic and finite nuclear mass theory). Today the so-
lution of the equation could be obtained with greater accuracy than reported
here. Imagine, that here the theory is limited by the precision of our knowledge
of the helium atom mass, which is “only” 12 significant figures.
• The effect of the non-zero size of the nucleus is small, it is practically never
taken into account in computations. If we enlarged the nucleus to the size of an
apple, the first Bohr orbit would be 10 km from the nucleus. And still (sticking
to our analogy) the electron is able to distinguish a point from an apple? Not
quite. It sees the (tiny) difference because the electron knows the region close
to the nucleus: it is there that it resides most often. Anyway the theory is able to
compute such a tiny effect.
41
C. Itzykson, J B. Zuber, “Quantum Field Theory”, McGraw-Hill, 1985, p. 193.
42
As remarked by Richard Feynman (see Additional Literature in the present chapter, p. 140) for
unknown reasons physics is based on the interaction of objects of spin
1
2
(like electrons or quarks)
mediated by objects of spin 1 (like photons, gluons or W particles).
43
With identical result, that increases enormously the confidence one may place in such results.
44
K. Pachucki, J. Sapirstein, Phys. Rev. A 63 (2001) 12504.
45
W. Cencek, K. Szalewicz, B. Jeziorski, Phys. Rev. Letters 86 (2001) 5675.
3.6 Beyond the Dirac equation
135
Table 3.1. Contributions of various physical effects (non-relativistic, Breit, QED and beyond QED) to the ionization energy and the dipole polarizability α
of the helium atom as well as comparison with the experimental values (all quantities in atomic units, i.e. e = 1,
¯
h =1, m
0
=1, where m
0
denotes the rest
mass of electron). The first column gives the symbol of the term in the Breit–Pauli Hamiltonian (3.70) as well as of the QED corrections given order by
order (first corresponding to the electron–positron vacuum polarization (QED), then, beyond quantum electrodynamics, to other particle–antiparticle pairs
(non-QED): μπ) split into several separate effects. The second column contains a short description of the effect. The estimated error (third column) is
given in parentheses in the units of the last figure reported
Term Physical interpretation Ionization energy [MHz] α [a.u.×10
−6
]
1
ˆ
H
0
Schrödinger equation 5 945 262288.62(4) 1 383 809.986(1)
δ non-zero size of the nucleus −29.55(4) 0.022(1)
ˆ
H
1
p
4
term 1 233 305.45(1) −987.88(1)
ˆ
H
2
(el-el) electron–electron retardation (Breit interaction) 4868488(1) −23219(1)
ˆ
H
2
(el-n) electron–nucleus retardation (Breit interaction) 31916(1) −0257(3)
ˆ
H
2
Breit interaction (total) 49 004.04(1) −23.476(3)
ˆ
H
3
spin–orbit 00
ˆ
H
4
(el-el) electron–electron Darwin term 11700883(1) −66083(1)
ˆ
H
4
(el-n) electron–nucleus Darwin term −1182 10099(1) 86485(2)
ˆ
H
4
Darwin term (total) −1 065 092.16(1) 798.77(2)
ˆ
H
5
spin–spin (total) −234 017.66(1) 132.166(1)
ˆ
H
6
spin-field 00
QED(c
−3
) vacuum polarization correction to electron–electron interaction −7248(1) 041(1)
QED(c
−3
) vacuum polarization correction to electron–nucleus interaction 146300(1) −1071(1)
QED(c
−3
) Total vacuum polarization in c
−3
order 139052(1) −1030(1)
QED(c
−3
) vac.pol. + other c
−3
QED correction −40 483.98(5) 30.66(1)
QED(c
−4
) vacuum polarization 1226(1) 0009(1)
QED(c
−4
) Total c
−4
QED correction −834.9(2) 0.56(22)
QED-h.o. Estimation of higher order QED correction 84(42) −0.06(6)
non-QED contribution of virtual muons, pions, etc. 0.05(1) −0.004(1)
Theory (total) 5 945 204 223(42)
2
1 383 760.79(23)
Experiment 5 945 204 238(45)
3
1 383 791(67)
4
1
G. Łach, B. Jeziorski, K. Szalewicz, Phys. Rev. Letters 92 (2004) 233001.
2
G.W.F. Drake, W.C. Martin, Can. J. Phys. 76 (1998) 679; V. Korobov, A. Yelkhovsky, Phys. Rev. Letters 87 (2001) 193003.
3
K.S.E. Eikema, W. Ubachs, W. Vassen, W. Hogervorst, Phys.Rev.A55 (1997) 1866.
4
F. Weinhold, J. Phys. Chem. 86 (1982) 1111.
